A HYPERSTABILITY RESULT FOR THE CAUCHY EQUATION

By Janusz Brzdek. Bulletin of the Australian Mathematical Society, 2013

88 citations recorded by the source corpus.

Abstract We prove a hyperstability result for the Cauchy functional equation $f(x+ y)= f(x)+ f(y)$, which complements some earlier stability outcomes of J. M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function $f$, mapping a normed space ${E}_{1} $ into a normed space ${E}_{2} $, and for all real numbers $r, s$ with $r+ s\gt 0$ one of the following two conditions must be valid: $$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = \infty , &&\displaystyle\end{eqnarray*}$$$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = 0. &&\displaystyle\end{eqnarray*}$$ In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/f12b0c3d320be05c1f7dccf435f6100be89269a1

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Citations in the corpus

Fixed Point Theory and the Ulam Stability (2014, background): “Rassias concerning optimality of estimations in Theorem 29) and in [69] (in proving stability of the equation ofp-Wright affine functions); in particular, it has been discovered in this way in [49] that the property of hyperstability for the additive Cauchy equation appears quite often in a natural ”
ON HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES (2015, unknown): “In essence, the method of proof was introduced in [6] and [7] and next used in [3, 5, 8, 18].”
Hyperstability and Superstability (2013, result): “Theorem 6 yields also the following two simple corollaries (see [21]), which correspond to some results from [26–33] on inhomogeneous Cauchy equation (18) and cocycle equation (19).”
Recent developments of the conditional stability of the homomorphism equation (2015, background): “, in [10, 23, 24, 25, 87] (for further references see [28]).”